The Voronoi Cells of the E6* and E7* Lattices

Edward PervinCarnegie-Mellon University

IN MEMORY OF MARSHALL HALL, JR.

Abstract

Recently, R. T. Worley succeeded in determining
the Voronoi cells of the E6* and E7* lattices. These turned out to be a 6-dimensional polytope with
720 vertices, and a 7-dimensional polytope with 576 vertices,
respectively. These two polytopes had been described in 1930
by H. S. M. Coxeter, who called them
the 0221 and the 123. They belong to a large class of
convex polytopes known as uniform polytopes. In this paper we
will use Wythoff's construction and a new result concerning the
lengths of edges of orthogonal trees to identify these polytopes.
We will also discuss the symmetry groups of these two
polytopes, which are of order 103,680 and 2,903,040 respectively.

Introduction

In the last five years, R. T. Worley [7,8] succeeded in
determining the Voronoi cells of the E6* and E7*
lattices. These turned out to be a 6-dimensional polytope with
720 vertices, and a 7-dimensional polytope with 576 vertices,
respectively. Worley was primarily interested in determining exact values
for the normalized second moments of these two polytopes. However, he
failed to mention that these
polytopes had been described as early as 1930 by H. S. M. Coxeter
[2, pp. 414-417],
who called them the 0221 and the
123. They belong to a large class of convex polytopes known as
uniform polytopes, and they can both be derived by a procedure called
Wythoff's construction.

In this paper we will independently identify these polytopes using
Wythoff's construction, aided by a new formula
for the lengths of the edges of a kind of simplex known as an orthogonal tree.
The chief virtue of this new derivation is that it does not require the
use of any coordinates whatsoever. We will also briefly discuss the symmetry
groups of these two polytopes, which are of order 103,680 and 2,903,040 respectively.

Wythoff's Construction and Coxeter Graphs

A polytope is uniform if all its facets are uniform and it is
``vertex-regular,'' that is, if it can be rotated so that any vertex
can be mapped onto any other vertex. To start the recursion off, a
polygon is defined to be uniform if it is regular.

In 1918, W. A. Wythoff [9] found almost all the uniform
polytopes in four dimensions by truncating the regular polytopes.
His method was generalized by Coxeter for higher dimensions
[3]. At the same time, Coxeter found a way to represent
uniform polytopes as graphs with certain nodes circled.
The uniform polytope represented by a diagram G is a cell of the
uniform polytope or honeycomb represented by the diagram H if and
only if G is a subdiagram of H.
The rest of this section is essentially a summary of relevant parts of
[3] and [4, chap. 11]; the reader
is referred there for details.

Coxeter found every example of a simplex that tiles either
Euclidean or spherical space by reflections. These simplices (called
fundamental simplices) have the property that any two bounding
hyperplanes meet each other at an angle which is a divisor of
180 degrees. An important example of a simplex that tiles
6-dimensional Euclidean space is shown in Figure 1.

Figure 1:
Six dimensional simplex.

Figure 2:
The E6 = 222 Lattice.

Each node of this graph represents a vertex of the simplex. If two
nodes share an edge then the two hyperplanes which are
opposite the corresponding vertices meet at an angle of 60 degrees;
otherwise the hyperplanes meet
at an angle of 90 degrees. (Since the graph is a tree,
this simplex is also called an orthogonal tree [5].)
An easy geometric argument shows that
each node can be labeled with an integer so that each node's label is
half the sum of its neighbors' labels. A vertex represented by a
node labeled 1 is called a special vertex. (The
subscripts are purely for ease of reference.)

Each node can also be interpreted as a reflection in the
corresponding bounding hyperplane. These seven reflections generate
an infinite group, sort of a 6-dimensional kaleidoscope. The image
of any special vertex, say 1a, under the action of this
reflection group is the lattice shown in Figure 2,
which is defined to be the lattice E6. (Hence Coxeter's
name 222. In general, if a graph has three branches of length p, q, and r, then pqr is obtained by
circling the last node of the branch of length p, and 0pqr
is obtained by circling the center node.) The images of any one of
the three special vertices 1a, 1b, or 1c
forms an E6 lattice. The lattice E6* consists
of the images of all three special vertices, and is therefore the
union of the three copies of E6:

The E6* lattice is not uniform since it requires
the union of more than one diagram to represent it, although we will
soon see that its Voronoi cell is uniform.

Finding the Voronoi Cells

To find the Voronoi cell of E6* we must find the point on
the simplex which is furthest from the three special vertices 1a,
1b, 1c. Formally, we want to find the
point P belonging to the simplex that maximizes

(1)

We will now show that P is in fact
the point labeled 3, thereby justifying:

Theorem 1
The Voronoi cell of E6* is 0221.

Proof: To find the lengths of the edges of the simplex, label each
edge of its graph with the inverse of the product of the labels of
that edge's two endpoints:

This new diagram is to be read as follows: to find the distance
between two vertices of the simplex, take the square root of the sum
of the numbers on the edges of the unique path between the two nodes
representing the two vertices. (This is the new formula mentioned
in the introduction. It was discovered by
the author in 1990, and then generalized to arbitrary orthogonal trees with angles
other than 60 and 90 degrees by Coxeter in [5].) For instance, the
distance from 1a to 1c is
.
This means that the triangles {1a, 2a, 3}, {1b, 2b, 3}, and {1c, 2c, 3} are all congruent
right triangles lying on three orthogonal planes.

So we see immediately that P = 3 is at least a local maximum
for equation (1) since any
infinitesimal adjustment would move P closer to some 1N. But since the images of vertex 3 form the honeycomb 0222,

which contains only one kind of cell,
namely the 0221 = 0212 = 0122,
the images of 3 must be the only vertices of the Voronoi cells
of E6* and 3 must be an absolute maximum for
equation (1). This
completes the proof of theorem 1.

The proof of the next theorem is very similar.

Theorem 2
The Voronoi cell of E7* is 123.

Proof: The graph for the fundamental simplex for the E7 and E7* lattices is:

Again, we have labeled each node with a number in such a way that
each node's label is half the sum of its neighbor's labels, and again
we have labeled each edge with the inverse of the product of the
labels of its endpoints. The rule for determining edge length is the
same as in theorem 1. The E7* lattice consists of the
images of the two special vertices 1x and 1y.
We want to find the point P which maximizes minN=x,ydist(P,1N).
This time, the desired
point P is easily seen to be the vertex 2z, the images
of which form the honeycomb 133,
which has only one kind of cell, the 123 = 132.
This completes the proof of theorem 2.

Rotation and Symmetry Groups

The rotation and symmetry groups of the 0221 and 123 and
the relationships between them are summarized in the following chart:

The groups in the diagram surrounded by thick boxes have a central
subgroup of order 2 containing the central inversion (which is a
rotation in even dimensions); those surrounded by thin
boxes do not. Indeed, each group surrounded by a thick box is the
direct product of the group to its lower left and the group of order
two containing the central inversion. Groups are connected by single
lines to normal subgroups, and by double lines to other subgroups.

The symmetry group of the 221 was originally studied in the
19th century as the group of automorphisms of the 27 lines on the cubic
surface, and the rotation group of the 123 was originally known as the
automorphism group of the 28 bitangents of a non-singular quartic
curve. Two good modern references for the study of these polytopes,
their groups, and their histories are [6, pp. 21-33] and
[1, pp. 26, 46].